Geometrical methods in hydrodynamics
Journées équations aux dérivées partielles (2001), article no. 2, 14 p.

We describe some recent results on a specific nonlinear hydrodynamical problem where the geometric approach gives insight into a variety of aspects.

@article{JEDP_2001____A2_0,
     author = {Constantin, Adrian},
     title = {Geometrical methods in hydrodynamics},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     publisher = {Universit\'e de Nantes},
     year = {2001},
     doi = {10.5802/jedp.586},
     zbl = {1007.35086},
     mrnumber = {1843403},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_2001____A2_0}
}
Constantin, Adrian. Geometrical methods in hydrodynamics. Journées équations aux dérivées partielles (2001), article  no. 2, 14 p. doi : 10.5802/jedp.586. http://www.numdam.org/item/JEDP_2001____A2_0/

[Ar] V. Arnold. Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble), 16:319-361, 1966. | Numdam | MR 202082 | Zbl 0148.45301

[Ar2] V. Arnold. Mathematical Methods of Classical Mechanics. Springer Verlag, New York, 1989. | MR 997295 | Zbl 0386.70001

[AK] V. Arnold and B. Khesin. Topological Methods in Hydrodynamics. Springer Verlag, New York, 1998. | MR 1612569 | Zbl 0902.76001

[BSS] R. Beals, D. Sattinger and J. Szmigielski. Multipeakons and a theorem of Stieltjes. Inverse Problems, 15:1-4, 1999. | MR 1675325 | Zbl 0923.35154

[Br] Y. Brenier. Minimal geodesics on groups of volume-preserving maps and generalized solutions to the Euler equations. Comm. Pure Appl. Math., 52:411-452, 1999. | MR 1658919 | Zbl 0910.35098

[CH] R. Camassa and D. Holm. An integrable shallow water equation with peaked solitons. Phys. Rev. Letters, 71:1661-1664, 1993. | MR 1234453 | Zbl 0972.35521

[AC] A. Constantin. A Lagrangian approximation to the water-wave problem. Appl. Math. Lett., to appear. | MR 1849230 | Zbl 0985.35069

[CE] A. Constantin and J. Escher. Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation. Comm. Pure Appl. Math., 51:475-504, 1998. | MR 1604278 | Zbl 0934.35153

[CK] A. Constantin and B. Kolev. On the geometric approach to the motion of inertial mechanical systems. Technical Report 6, Lund University, 2001. | MR 1930889

[CM] A. Constantin and H. P. Mckean. A shallow water equation on the circle. Comm. Pure Appl. Math., 52:949-982, 1999. | MR 1686969 | Zbl 0940.35177

[CMo] A. Constantin and L. Molinet. Global weak solutions for a shallow water equation. Comm. Math. Phys., 211:45-61, 2000. | MR 1757005 | Zbl 1002.35101

[CS] A. Constantin and W. Strauss. Stability of peakons. Comm. Pure Appl. Math., 53:603-610, 2000. | MR 1737505 | Zbl 1049.35149

[EM] D. Ebin and J. E. Marsden. Groups of diffeomorphisms and the notion of an incompressible fluid. Ann. of Math., 92:102-163, 1970. | MR 271984 | Zbl 0211.57401

[FF] A. S. Fokas and B. Fuchssteiner. Symplectic structures, their Bäcklund transformation and hereditary symmetries. Physica D, 4:47-66, 1981. | MR 636470

[Ha] R. Hamilton. The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc., 7:65-222, 1982. | MR 656198 | Zbl 0499.58003

[HM] A. Himonas and G. Misiolek. The Cauchy problem for an integrable shallow-water equation. Differential and Integral Equations, 14:821-831, 2001. | MR 1828326 | Zbl 1009.35075

[Jo] R. S. Johnson. A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, 1997. | MR 1629555 | Zbl 0892.76001

[La] S. Lang. Fundamentals of Differential Geometry. Springer Verlag, New York, 1999. | MR 1666820 | Zbl 0932.53001

[LO] Yi Li and P. Olver. Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differential Equations, 162:27-63, 2000. | MR 1741872 | Zbl 0958.35119

[McK] H. P. Mckean. Breakdown of a shallow water equation. Asian J. Math., 2:203-208, 1998. | MR 1734131 | Zbl 0959.35140

[Mil] J. Milnor. Remarks on infinite-dimensional Lie groups. In Relativity, Groups and Topology (Les Houches, 1983), pages 1009-1057. North-Holland, Amsterdam, 1984. | MR 830252 | Zbl 0594.22009

[Mi] G. Misiolek. A shallow water equation as a geodesic flow on the Bott-Virasoro group. J. Geom. Phys., 24:203-208, 1998. | MR 1491553 | Zbl 0901.58022

[Ol] P. Olver. Applications of Lie Groups to Differential Equations. Springer Verlag, New York, 1993. | MR 1240056 | Zbl 0785.58003

[Sh] A. Shnirelman. Generalized fluid flows, their approximation and applications. Geom. Funct. Anal., 4:586-620, 1994. | MR 1296569 | Zbl 0851.76003

[Wu]S. Wu. Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc., 12:445-495, 1999. | MR 1641609 | Zbl 0921.76017